Given a surface \(\Sigma _{n,g}\)  of genus \(g\)  and \(n\)  boundary circles (or \(n\)  punctures), and a complex reductive or compact Lie group \(G\) , the moduli space of representations of the fundamental group \(\pi _1(\Sigma _{n,g})\)  into \(G\) , the \(G\) -character variety of \(\pi _1(\Sigma _{n,g})\), has a natural Poisson structure. This structure was given by Goldman [1]} in the closed case, and extended in [2]} to the case of surfaces with boundary (see also [3]}). The representation space also has an equivalent interpretation as a space of flat connections, and from this point-of-view one can define the Poisson structure with an approach pioneered by Atiyah
and Bott [4]} when the surface is compact, and extended by Jeffrey in [5]} to punctured surfaces (see also [3]}). In this paper we consider the effect of mappings between surfaces. Given an appropriate mapping between two surfaces \(f\,:\,\Sigma _1\,\longrightarrow \,\Sigma _2\) , there is a natural morphism between their character varieties:
\(\Phi \, :\, \mathfrak {X}_{\Sigma _2}(G) \,\longrightarrow \, \mathfrak {X}_{\Sigma _1}(G) \, .\) 
